integral de (sqrt(36+(ln(2x))^2))/x

Solución

\int \frac{36 + ( ln ( 2 x ) ) ^{2}}{x} d x

36 (\frac{1}{72} ln (2 x) 36 + ln^{2} (2 x) + \frac{1}{2} ln (\frac{1}{6} ln (2 x) + 36 + ln^{2} (2 x))) + C

Pasos de solución

\int \frac{36 + ( ln ( 2 x ) ) ^{2}}{x} d x

Aplicar integración por sustitución

: \int \frac{36 + ln ^{2} ( u )}{u} d u

= \int \frac{36 + ln ^{2} ( u )}{u} d u

Aplicar integración por sustitución

: \int 36 + v^{2} d v

= \int 36 + v^{2} d v

Aplicar integración por sustitución trigonométrica

: \int 36 sec^{3} (w) d w

= \int 36 sec^{3} (w) d w

Sacar la constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 36 \cdot \int sec^{3} (w) d w

Aplicar la reducción de integrales

= 36 (\frac{sec ^{2} ( w ) sin ( w )}{2} + \frac{1}{2} \cdot \int sec (w) d w)

\int sec (w) d w = ln ∣ tan (w) + sec (w) ∣

= 36 (\frac{sec ^{2} ( w ) sin ( w )}{2} + \frac{1}{2} ln ∣ tan (w) + sec (w) ∣)

Sustitución hacia atrás

= 36 (\frac{sec ^{2} ( arctan ( \frac{1}{6} ln ( 2 x ) ) ) sin ( arctan ( \frac{1}{6} ln ( 2 x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{6} ln (2 x))) + sec (arctan (\frac{1}{6} ln (2 x))))

Simplificar

36 (\frac{sec ^{2} ( arctan ( \frac{1}{6} ln ( 2 x ) ) ) sin ( arctan ( \frac{1}{6} ln ( 2 x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{6} ln (2 x))) + sec (arctan (\frac{1}{6} ln (2 x)))) : 36 (\frac{1}{72} ln (2 x) 36 + ln^{2} (2 x) + \frac{1}{2} ln (\frac{1}{6} ln (2 x) + 36 + ln^{2} (2 x)))

= 36 (\frac{1}{72} ln (2 x) 36 + ln^{2} (2 x) + \frac{1}{2} ln (\frac{1}{6} ln (2 x) + 36 + ln^{2} (2 x)))

Agregar una constante a la solución

= 36 (\frac{1}{72} ln (2 x) 36 + ln^{2} (2 x) + \frac{1}{2} ln (\frac{1}{6} ln (2 x) + 36 + ln^{2} (2 x))) + C

\frac{d}{d x} (3 y) = 5

4 x e^{x}

\int sin (t^{3}) d t

f (x) = \frac{1 + x - 4 x}{x}

x \cdot sin (\frac{2}{x})